Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (5): 1290-1308.doi: 10.1007/s10473-019-0508-8

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GLOBAL EXISTENCE, EXPONENTIAL DECAY AND BLOW-UP IN FINITE TIME FOR A CLASS OF FINITELY DEGENERATE SEMILINEAR PARABOLIC EQUATIONS

Hua CHEN, Huiyang XU   

  1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China
  • Received:2018-02-06 Revised:2019-01-08 Online:2019-10-25 Published:2019-11-11
  • Contact: Hua CHEN E-mail:chenhua@whu.edu.cn
  • Supported by:
    This work was supported by National Natural Science Foundation of China (11631011 and 11626251).

Abstract: In this paper, we study the initial-boundary value problem for the semilinear parabolic equations ut -△Xu=|u|p-1u, where X=(X1, X2, …, Xm) is a system of real smooth vector fields which satisfy the H?rmander's condition, and is a finitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Finally, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy, and also we discuss the asymptotic behavior of the global solutions.

Key words: finitely degenerate parabolic equation, global existence, blow-up, decay estimate

CLC Number: 

  • 35K58
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